TITLE: Is all of mathematics set theory in disguise?
QUESTION [19 upvotes]: Is all of mathematics, just set theory in disguise? We know the ZFC axioms allow us to encode mathematical objects as sets of some kind. So, even theorems about, say, Hilbert spaces can be written down, tediously, as formulas of ZFC. So, does this mean all of mathematics is set theory in disguise?

REPLY [13 votes]: I would say that it's the other way round: most mathematical topics can be disguised in set theory, but their true nature is only revealed when the disguise is cast off. For a simple example, the set theoretic view of the permutation groups $S_n$ as a set of functions each function being represented as a set of pairs doesn't help much in elementary group theory.